Why Build a Discrete Price Menu?
A customer-level renewal decision may allow many possible prices within a permitted range, while an evaluation or decision process needs a finite set of candidates. A discrete price menu is that finite set: the prices available for a particular policy to be evaluated. Its construction determines which alternatives can be compared. A price that is not in the menu is not available to that evaluation, even if it falls between permitted limits.
Menus can be specified as dollar premiums or as percentage changes from a policy's current premium. These choices are not interchangeable: equal dollar increments represent different percentage changes for policies with different starting premiums. In either case, the process is to define the starting premium, the permitted price range, the grid spacing, and the rule for including endpoints. Then construct the candidates and verify that each one meets the stated conditions.
Constructing a Menu From Dollar Increments
For a dollar grid, choose a starting value \(P_{\min}\), an increment \(\Delta>0\), and an upper bound \(P_{\max}\). The candidates are \(P_{\min}+k\Delta\) for nonnegative integers \(k\) that keep the candidate within the specified range. If the upper bound is not reached by a grid point, the menu does not automatically include that bound: whether to add it must be part of the menu rule.
A regular grid makes the spacing between interior candidates transparent, but bounds and spacing must be compatible. For example, a range from $700 to $900 with $60 increments reaches $880 but not $900. If $900 is expressly required as an available endpoint, a rule is needed to include it; doing so creates a final interval of $20 rather than $60.
Worked Example: A Dollar-Increment Menu With an Endpoint Check
A hypothetical policy has a current annual premium of $820. A proposed menu uses candidate premiums from $700 through $900, inclusive, in $50 increments from the lower bound. Construct the menu, determine the percentage change from the current premium for each candidate, and check whether $900 is included.
Method. Start at $700 and add $50 repeatedly while the candidate does not exceed $900:
Thus the menu is \(\{700,750,800,850,900\}\). The percentage change for each candidate is \((P-P_0)/P_0\), where \(P_0=820\). The calculations are:
- At $700: \((700-820)/820=-120/820=-0.146341\ldots\), or \(-14.6341\ldots\%\).
- At $750: \((750-820)/820=-70/820=-0.085365\ldots\), or \(-8.5365\ldots\%\).
- At $800: \((800-820)/820=-20/820=-0.024390\ldots\), or \(-2.4390\ldots\%\).
- At $850: \((850-820)/820=30/820=0.036585\ldots\), or \(3.6585\ldots\%\).
- At $900: \((900-820)/820=80/820=0.097561\ldots\), or \(9.7561\ldots\%\).
The increments land exactly on both bounds, so the menu includes $900 and every adjacent difference is $50. The percentage changes are rounded to four decimal places; the dollar candidates are exact. Common mistake avoided: stopping at $850 because it is the last candidate strictly below $900. The stated rule includes the upper endpoint, and $900 is reached exactly.
Constructing a Menu From Percentage Changes
A percentage-change grid is built relative to each policy's own current premium. If the permitted change rates are \(r_{\min}, r_{\min}+\Delta r,\ldots,r_{\max}\), each candidate premium is \(P_0(1+r)\). The rate \(r\) is expressed as a decimal in this formula: for example, a 4% increase means \(r=0.04\). A percentage grid therefore preserves the same relative changes across policies but generally produces different dollar increments.
Price limits may remove some candidates from the grid. Apply the limits to the resulting premiums, not to the change rates unless the limits are themselves stated as rate limits. Also verify the order of operations: first convert each rate to its candidate premium, then retain the premiums allowed by the stated dollar bounds.
Worked Example: Percentage Grid Clipped by Premium Limits
A hypothetical policy renews at $1,250. Candidate rate changes are \(-8\%\), \(-4\%\), \(0\%\), \(4\%\), and \(8\%\). The permitted renewal premium range is $1,180 to $1,330, inclusive. Construct the available menu and state which candidate rates remain.
Method. Convert each rate \(r\) to a premium using \(P=1{,}250(1+r)\):
- \(-8\%\): \(1{,}250(1-0.08)=1{,}250(0.92)=\$1{,}150\).
- \(-4\%\): \(1{,}250(1-0.04)=1{,}250(0.96)=\$1{,}200\).
- \(0\%\): \(1{,}250(1+0)=\$1{,}250\).
- \(4\%\): \(1{,}250(1+0.04)=1{,}250(1.04)=\$1{,}300\).
- \(8\%\): \(1{,}250(1+0.08)=1{,}250(1.08)=\$1{,}350\).
Now apply the inclusive limits \(1{,}180\leq P\leq1{,}330\). The $1,150 candidate is below the minimum, and the $1,350 candidate is above the maximum. The remaining menu is \(\{\$1{,}200,\ \$1{,}250,\ \$1{,}300\}\), corresponding to \(-4\%,0\%,4\%\). Each retained value is within the range, and each excluded value violates one bound. Common mistake avoided: treating the allowed premium range as a rate range. The limits are in dollars, so they must be checked against the converted premiums.
How Spacing and Bounds Affect Available Choices
Menu spacing controls resolution. A coarse grid is shorter and simpler to evaluate, but may omit prices near an attractive price level. A finer grid adds choices and can represent the permitted range more closely, at the cost of evaluating more candidates. Bounds have a separate effect: they exclude prices outside the allowed interval, while endpoint conventions determine whether a price exactly on a limit is available.
The effect of spacing can be stated precisely for a regular grid. Suppose the relevant interval is \([L,U]\), both endpoints are included, and the grid divides the interval into equal steps of size \(h\). Every price in the interval is within \(h/2\) of at least one grid point. To see why, take any price between two adjacent grid points. Its distance to the nearer endpoint of that subinterval cannot exceed half the subinterval length \(h\). Consequently, reducing the step from \(h\) to \(h/2\) reduces this maximum distance from \(h/2\) to \(h/4\), provided the refined grid and endpoint rules are implemented as stated.
This is a statement about price resolution, not a guarantee about the financial value of the chosen price. If an ideal price lies between candidates, the nearest menu price may differ from it; how consequential that difference is depends on how expected contribution or another evaluation measure varies with price. The menu construction itself does not establish that the nearest candidate is best.
Worked Example: Comparing Percentage-Grid Resolution Across Policies
Two hypothetical policies have current premiums of $600 and $1,200. For each, construct candidate prices using rate changes of \(-10\%\), \(-5\%\), \(0\%\), \(5\%\), and \(10\%\), then apply inclusive premium limits of $550 to $1,300. Compare the dollar spacing for the two policies and identify which candidates are removed.
Method. For the $600 policy, the five premiums are:
- \(600(0.90)=\$540\).
- \(600(0.95)=\$570\).
- \(600(1.00)=\$600\).
- \(600(1.05)=\$630\).
- \(600(1.10)=\$660\).
Applying the $550 minimum removes $540; the $660 maximum does not remove any other candidate. Its available menu is \(\{\$570,\$600,\$630,\$660\}\), with $30 spacing between adjacent candidates.
For the $1,200 policy, the five premiums are:
- \(1{,}200(0.90)=\$1{,}080\).
- \(1{,}200(0.95)=\$1{,}140\).
- \(1{,}200(1.00)=\$1{,}200\).
- \(1{,}200(1.05)=\$1{,}260\).
- \(1{,}200(1.10)=\$1{,}320\).
The $1,080 and $1,320 candidates violate the lower and upper limits, respectively. Its available menu is \(\{\$1{,}140,\$1{,}200,\$1{,}260\}\), with $60 spacing between adjacent candidates. Both policies use identical rate changes, but the second policy's dollar spacing is twice as large because its base premium is twice as large. Each remaining candidate is within the common dollar limits. Common mistake avoided: assuming equal percentage steps produce equal dollar steps across policies.
A Systematic Construction and Validation Process
A menu should be reproducible: another analyst given the same base premium, range, spacing, and endpoint rule should obtain the same candidates. The following checks help identify errors before the menu is used to evaluate renewal decisions.
Identify whether candidates are specified as dollar premiums or percentage changes, and record the policy's current premium if changes are relative to it.
Record the minimum and maximum, the increment, and whether each endpoint must be included. For a percentage grid, specify whether the bounds are dollar premiums or rate changes.
For a dollar grid, add the stated dollar increment. For a percentage grid, convert each rate with \(P=P_0(1+r)\).
Remove candidates outside the permitted range. Check that retained values meet the limits, required endpoints are present, and the menu is nonempty.
A regular grid may fail to land on an endpoint. If exact endpoint inclusion is required, the menu rule must explain how to handle the remaining interval. Options include adding the endpoint as a final candidate, which makes the last step shorter, or selecting an increment that divides the interval. Do not silently extend beyond the bound or assume that the nearest grid point is an endpoint.
A menu can also be empty after applying limits. For instance, if all candidates from a percentage grid convert to premiums outside a narrow dollar range, no candidate remains. The correct conclusion is that the specified grid and limits produce no available price; it is not appropriate to insert a price without a stated rule. Menu feasibility and the reasons for exclusions should be clear before any candidate is evaluated.
Common Mistakes and Practical Interpretation
- Mixing percentage points and dollars. A 5% step is a relative change, not a fixed dollar increment; convert it using the policy's base premium.
- Forgetting the base premium. A percentage-change menu cannot be converted to prices without \(P_0\).
- Including an endpoint by assumption. A bound is not necessarily a grid point. Check the arithmetic and follow the stated endpoint rule.
- Applying limits to the wrong quantity. If limits are dollar premiums, convert rate candidates to dollars before checking them.
- Equating a finer menu with a better decision. A finer grid offers more candidates, but the selected price still depends on the evaluation method and its inputs.
The practical purpose of a discrete menu is to make the set of alternatives explicit and manageable. Its definition should preserve the intended price range, provide suitable resolution, and allow each candidate to be traced back to its grid rule. These construction principles are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, construct or check the candidate menu before reviewing the answer.
- A hypothetical policy renews at $900. Its dollar menu starts at $780 and adds $40 through a maximum of $900, inclusive. List the menu and state whether the maximum is reached. (Answer: \(\{\$780,\$820,\$860,\$900\}\); yes, the upper bound is reached.)
- A policy has a current premium of $800 and a rate-change grid of \(-6\%, -3\%, 0\%, 3\%, 6\%\). What are the unfiltered candidate premiums? (Answer: \(\{\$752,\$776,\$800,\$824,\$848\}\).)
- A $1,500 policy uses rate changes of \(-10\%, -5\%, 0\%, 5\%, 10\%\), with inclusive premium limits of $1,400 to $1,600. What menu remains? (Answer: \(\{\$1{,}425,\$1{,}500,\$1{,}575\}\); the $1,350 and $1,650 candidates are excluded.)
- A dollar grid runs from $1,000 to $1,100 in $30 steps, with both endpoints required. What regular-grid candidates are generated from the lower bound, and what endpoint issue arises? (Answer: \(\{\$1{,}000,\$1{,}030,\$1{,}060,\$1{,}090\}\); $1,100 is not reached, so it must be added by an explicit endpoint rule or the increment changed.)
- Two policies have current premiums of $500 and $1,000 and use the same \(-4\%,0\%,4\%\) grid. What are their candidate menus, and how do their dollar increments compare? (Answer: \(\{\$480,\$500,\$520\}\) and \(\{\$960,\$1{,}000,\$1{,}040\}\); the second policy's adjacent dollar increments are twice as large.)